A Remarkable Identity Involving Bessel Functions

TL;DR

This paper introduces a novel, simple identity involving integrals and sums of Bessel functions, enabling efficient evaluation of certain integrals with practical numerical benefits.

Contribution

It presents a new, elegant identity linking integrals and sums of Bessel functions, enhancing computational methods for these special functions.

Findings

The identity simplifies evaluation of Bessel function integrals.

Numerical examples demonstrate fast convergence.

The identity is applicable to a wide range of cases.

Abstract

We consider a new identity involving integrals and sums of Bessel functions. The identity provides new ways to evaluate integrals of products of two Bessel functions. The identity is remarkably simple and powerful since the summand and integrand are of exactly the same form and the sum converges to the integral relatively fast for most cases. A proof and numerical examples of the identity are discussed.